Lump And Mixed Rogue-Soliton Solutions Of The (2+1) Dimensional Soliton Equations

During the development of nonlinear systems,the study of lump and rogue wave solutions has drawn more and more attention.The special structures and potential destructiveness of lump wave and rogue wave are the focus of attention.Especially rogue wave,rogue wave is a disaster on the ocean.It is difficult to be predicted and controlled when it happen.But the research on the propagation of optical rogue wave in optical fiber systems have great values for optical fiber communication.In addition,the rogue wave are also widely used in plasma systems.In the actual physical background,whether in the marine system or in the optical system,the freedom of space and time can not be treated separately.Therefore,how to construct the(2+1)-dimensional coupled complex equations’ solutions,such as lump and mixed rogue-soliton solutions,is an especially intriguing topic.This paper mainly studies the exact solutions of two different systems based on the direct construction of bilinear properties,the Mel’nikov system and the two component Maccari system.The thesis is divided into four parts,which are the origin and current research status of the lump wave and the rogue wave,the exact solutions of the two component Mel’nikov system and the Maccari system respectively,and the summary and prospect of the paper.The main results of this thesis are as follows:(1)By constructing a rational function on the Hirota bilinear form of the Mel’nikov system,the lump solution and line rogue wave are obtained.By combining a rational function and different exponential functions,mixed solutions between the lump and soliton are derived.These solutions describe the interaction phenomena of the lump-bright soliton with fission and fusion,the half-line rogue wave with a bright soliton,and a rogue wave excited from the bright soliton pair,respectively.When the trigonometric and exponential functions are linearly combined,the homoclinic breather solution is obtained.The rogue wave has three different basic forms,the line rogue wave,the half-line rogue wave and therational rogue wave.When we choose different construction methods,the peaks of these rogue waves will appear at different times.(2)By constructing a rational function on the Hirota bilinear form of the two component Maccari system,the lump solution and line rogue wave are obtained.By combining a rational function and different exponential functions,the interaction phenomena of the lump-soliton with fission and fusion,and the rogue wave with soliton,respectively.Real scalar fields are bright solitons,while complex scalar fields are dark solitons.The rogue wave has four different basic forms,the line rogue wave,the half-line rogue wave,the Kink rogue wave and the rational rogue wave.When we choose different construction methods,the peaks of these rogue waves will appear at different times.